Step |
Hyp |
Ref |
Expression |
1 |
|
caurcvg.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
caurcvg.3 |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ ) |
3 |
|
caurcvg.4 |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) |
4 |
|
uzssz |
⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℤ |
5 |
1 4
|
eqsstri |
⊢ 𝑍 ⊆ ℤ |
6 |
|
zssre |
⊢ ℤ ⊆ ℝ |
7 |
5 6
|
sstri |
⊢ 𝑍 ⊆ ℝ |
8 |
7
|
a1i |
⊢ ( 𝜑 → 𝑍 ⊆ ℝ ) |
9 |
|
1rp |
⊢ 1 ∈ ℝ+ |
10 |
9
|
ne0ii |
⊢ ℝ+ ≠ ∅ |
11 |
|
r19.2z |
⊢ ( ( ℝ+ ≠ ∅ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) → ∃ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) |
12 |
10 3 11
|
sylancr |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) |
13 |
|
eluzel2 |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ ) |
14 |
13 1
|
eleq2s |
⊢ ( 𝑚 ∈ 𝑍 → 𝑀 ∈ ℤ ) |
15 |
1
|
uzsup |
⊢ ( 𝑀 ∈ ℤ → sup ( 𝑍 , ℝ* , < ) = +∞ ) |
16 |
14 15
|
syl |
⊢ ( 𝑚 ∈ 𝑍 → sup ( 𝑍 , ℝ* , < ) = +∞ ) |
17 |
16
|
a1d |
⊢ ( 𝑚 ∈ 𝑍 → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 → sup ( 𝑍 , ℝ* , < ) = +∞ ) ) |
18 |
17
|
rexlimiv |
⊢ ( ∃ 𝑚 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 → sup ( 𝑍 , ℝ* , < ) = +∞ ) |
19 |
18
|
rexlimivw |
⊢ ( ∃ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 → sup ( 𝑍 , ℝ* , < ) = +∞ ) |
20 |
12 19
|
syl |
⊢ ( 𝜑 → sup ( 𝑍 , ℝ* , < ) = +∞ ) |
21 |
5
|
sseli |
⊢ ( 𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ ) |
22 |
5
|
sseli |
⊢ ( 𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ ) |
23 |
|
eluz |
⊢ ( ( 𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ↔ 𝑚 ≤ 𝑘 ) ) |
24 |
21 22 23
|
syl2an |
⊢ ( ( 𝑚 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍 ) → ( 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ↔ 𝑚 ≤ 𝑘 ) ) |
25 |
24
|
biimprd |
⊢ ( ( 𝑚 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍 ) → ( 𝑚 ≤ 𝑘 → 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) |
26 |
25
|
expimpd |
⊢ ( 𝑚 ∈ 𝑍 → ( ( 𝑘 ∈ 𝑍 ∧ 𝑚 ≤ 𝑘 ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ) ) |
27 |
26
|
imim1d |
⊢ ( 𝑚 ∈ 𝑍 → ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) → ( ( 𝑘 ∈ 𝑍 ∧ 𝑚 ≤ 𝑘 ) → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
28 |
27
|
exp4a |
⊢ ( 𝑚 ∈ 𝑍 → ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) → ( 𝑘 ∈ 𝑍 → ( 𝑚 ≤ 𝑘 → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) ) ) |
29 |
28
|
ralimdv2 |
⊢ ( 𝑚 ∈ 𝑍 → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 → ∀ 𝑘 ∈ 𝑍 ( 𝑚 ≤ 𝑘 → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) ) |
30 |
29
|
reximia |
⊢ ( ∃ 𝑚 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 → ∃ 𝑚 ∈ 𝑍 ∀ 𝑘 ∈ 𝑍 ( 𝑚 ≤ 𝑘 → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
31 |
30
|
ralimi |
⊢ ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑘 ∈ 𝑍 ( 𝑚 ≤ 𝑘 → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
32 |
3 31
|
syl |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑘 ∈ 𝑍 ( 𝑚 ≤ 𝑘 → ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 ) ) |
33 |
8 2 20 32
|
caurcvgr |
⊢ ( 𝜑 → 𝐹 ⇝𝑟 ( lim sup ‘ 𝐹 ) ) |
34 |
14
|
a1d |
⊢ ( 𝑚 ∈ 𝑍 → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 → 𝑀 ∈ ℤ ) ) |
35 |
34
|
rexlimiv |
⊢ ( ∃ 𝑚 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 → 𝑀 ∈ ℤ ) |
36 |
35
|
rexlimivw |
⊢ ( ∃ 𝑥 ∈ ℝ+ ∃ 𝑚 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑚 ) ( abs ‘ ( ( 𝐹 ‘ 𝑘 ) − ( 𝐹 ‘ 𝑚 ) ) ) < 𝑥 → 𝑀 ∈ ℤ ) |
37 |
12 36
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
38 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
39 |
|
fss |
⊢ ( ( 𝐹 : 𝑍 ⟶ ℝ ∧ ℝ ⊆ ℂ ) → 𝐹 : 𝑍 ⟶ ℂ ) |
40 |
2 38 39
|
sylancl |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℂ ) |
41 |
1 37 40
|
rlimclim |
⊢ ( 𝜑 → ( 𝐹 ⇝𝑟 ( lim sup ‘ 𝐹 ) ↔ 𝐹 ⇝ ( lim sup ‘ 𝐹 ) ) ) |
42 |
33 41
|
mpbid |
⊢ ( 𝜑 → 𝐹 ⇝ ( lim sup ‘ 𝐹 ) ) |